The melting ice cream can be a number of things.
Check that the fan in the freezer is running.
Check that the condenser coils at the back/bottom are clean and the fan down there is working.
Could be that you have a gasket leak around the freezer door but usually you would see frost forming inside the freezer.
Could also be the thermostat is shot, best to get a thermometer and measure the freezer temperature. Note that there is a fairly long lag between when you set the thermostat and when you actually see a temperature difference in the freezer. Depends on how full the freezer is.